## Twisted Alexander polynomials - an overview

Stefan Friedl

September 2010

Questions about knots

By a knotK we mean a closed embedded curve inS^{3}.

We list some goals in knot theory. (2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot (3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not. (5) Determine which knots are periodic. (6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

## Questions about knots

By a knotK we mean a closed embedded curve inS^{3}.
We list some goals in knot theory.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot (3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not. (5) Determine which knots are periodic. (6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

Questions about knots

By a knotK we mean a closed embedded curve inS^{3}.
We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot (3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not. (5) Determine which knots are periodic. (6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

## Questions about knots

By a knotK we mean a closed embedded curve inS^{3}.
We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Any knotK bounds an orientable embedded surface (Seifert surface).

(2) Determine the genus g(K) of K. (2’) Determine the free genus of a given knot (3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not. (5) Determine which knots are periodic. (6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

## Questions about knots

By a knotK we mean a closed embedded curve inS^{3}.
We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Any knotK bounds an orientable embedded surface (Seifert surface). Thegenus of K is the minimal genus among all Seifert surfaces.

(2) Determine the genusg(K) ofK.

## Questions about knots

By a knotK we mean a closed embedded curve inS^{3}.
We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Any knotK bounds an orientable embedded surface (Seifert surface). Thegenus of K is the minimal genus among all Seifert surfaces.

Goal: determine the genusg(K) of a given knot

(2) Determine the genusg(K) of K.

## Questions about knots

By a knotK we mean a closed embedded curve inS^{3}.
We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

## Questions about knots

By a knotK we mean a closed embedded curve inS^{3}.
We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Any knot admits a Seifert surface Σ such thatπ_{1}(S^{3}\Σ) is
free.

## Questions about knots

By a knotK we mean a closed embedded curve inS^{3}.
We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Any knot admits a Seifert surface Σ such thatπ_{1}(S^{3}\Σ) is
free. The minimal genus of such a Seifert surface is thefree genus
ofK.

## Questions about knots

By a knotK we mean a closed embedded curve inS^{3}.
We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Any knot admits a Seifert surface Σ such thatπ_{1}(S^{3}\Σ) is
free. The minimal genus of such a Seifert surface is thefree genus
ofK.

Goal: determine the free genusg_{free}(K) of a knot.

(2’) Determine the free genus of a given knot

(3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not. (5) Determine which knots are periodic. (6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

## Questions about knots

By a knotK we mean a closed embedded curve inS^{3}.
We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot

(3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not. (5) Determine which knots are periodic. (6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

## Questions about knots

By a knotK we mean a closed embedded curve inS^{3}.
We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot

(3) A knot isfibered if there exists a fibrationS^{3}\K →S^{1}

(3) Determine whether a given knot is fibered

(4) Determine whetherK is slice or not. (5) Determine which knots are periodic. (6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

## Questions about knots

By a knotK we mean a closed embedded curve inS^{3}.
We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot

(3) A knot isfibered if there exists a fibrationS^{3}\K →S^{1} (i.e. a
map such that the preimage of an interval is a surface times an
interval).

(3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not.

(5) Determine which knots are periodic. (6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

## Questions about knots

By a knotK we mean a closed embedded curve inS^{3}.
We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot

(3) A knot isfibered if there exists a fibrationS^{3}\K →S^{1} (i.e. a
map such that the preimage of an interval is a surface times an
interval). Note that a fiber is a genus minimizing Seifert surface.

(3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not. (5) Determine which knots are periodic. (6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

## Questions about knots

By a knotK we mean a closed embedded curve inS^{3}.
We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot

(3) A knot isfibered if there exists a fibrationS^{3}\K →S^{1} (i.e. a
map such that the preimage of an interval is a surface times an
interval). Note that a fiber is a genus minimizing Seifert surface.

Goal: determine whether a knotK is fibered.

## Questions about knots

By a knotK we mean a closed embedded curve inS^{3}.
We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot (3) Determine whether a given knot is fibered

(4) Determine whetherK is slice or not. (5) Determine which knots are periodic. (6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

## Questions about knots

By a knotK we mean a closed embedded curve inS^{3}.
We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot
(3) Determine whether a given knot is fibered
(4) A knot issliceif it bounds a smooth disk in D^{4}.

(4) Determine whetherK is slice or not. (5) Determine which knots are periodic. (6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

## Questions about knots

By a knotK we mean a closed embedded curve inS^{3}.
We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot
(3) Determine whether a given knot is fibered
(4) A knot issliceif it bounds a smooth disk in D^{4}.
Goal: determine which knots are slice.

(4) Determine whether K is slice or not.

(5) Determine which knots are periodic. (6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

## Questions about knots

By a knotK we mean a closed embedded curve inS^{3}.
We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot (3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not.

(5) Determine which knots are periodic. (6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

## Questions about knots

By a knotK we mean a closed embedded curve inS^{3}.
We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot (3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not.

(5) A knotK is periodic of order n

(5) Determine which knots are periodic.

(6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

## Questions about knots

By a knotK we mean a closed embedded curve inS^{3}.
We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(2’) Determine the free genus of a given knot (3) Determine whether a given knot is fibered (4) Determine whetherK is slice or not.

(5) A knotK is periodic of ordernif there exists a homeomorphism
ofS^{3} of order r

(5) Determine which knots are periodic. (6) Determine which knots are amphichiral.

(7) Determine the partial order≥of knots.

## Questions about knots

By a knotK we mean a closed embedded curve inS^{3}.
We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(5) A knotK is periodic of ordernif there exists a homeomorphism
ofS^{3} of order r which fixes an unknot pointwise andK setwise.

## Questions about knots

By a knotK we mean a closed embedded curve inS^{3}.
We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(5) A knotK is periodic of ordernif there exists a homeomorphism
ofS^{3} of order r which fixes an unknot pointwise andK setwise.

Goal: Determine which knots are periodic.

(5) Determine which knots are periodic.

(6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

## Questions about knots

By a knotK we mean a closed embedded curve inS^{3}.
We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(5) Determine which knots are periodic.

(6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

## Questions about knots

By a knotK we mean a closed embedded curve inS^{3}.
We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(5) Determine which knots are periodic.

(6) Given a knotK denote by K^{∗} its mirror image

(6) Determine which knots are amphichiral.

(7) Determine the partial order≥of knots.

## Questions about knots

By a knotK we mean a closed embedded curve inS^{3}.
We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(5) Determine which knots are periodic.

(6) Given a knotK denote by K^{∗} itsmirror imagei.e. the result of
reflectingK in a plane.

(6) Determine which knots are amphichiral.

(7) Determine the partial order≥of knots.

## Questions about knots

By a knotK we mean a closed embedded curve inS^{3}.
We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(5) Determine which knots are periodic.

(6) Given a knotK denote by K^{∗} itsmirror imagei.e. the result of
reflectingK in a plane. A knot which equals its mirror image is
calledamphichiral.

(6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

## Questions about knots

By a knotK we mean a closed embedded curve inS^{3}.
We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(5) Determine which knots are periodic.

(6) Given a knotK denote by K^{∗} itsmirror imagei.e. the result of
reflectingK in a plane. A knot which equals its mirror image is
calledamphichiral. Goal: Determine which knots are amphichiral.

(6) Determine which knots are amphichiral. (7) Determine the partial order≥of knots.

## Questions about knots

By a knotK we mean a closed embedded curve inS^{3}.
We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(5) Determine which knots are periodic.

(6) Determine which knots are amphichiral.

(7) Determine the partial order≥of knots.

## Questions about knots

By a knotK we mean a closed embedded curve inS^{3}.
We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(5) Determine which knots are periodic.

(6) Determine which knots are amphichiral.

(7) We writeK1 ≥K2 if there exists an epimorphism
π1(S^{3}\K1)→π1(S^{3}\K2).

(7) Determine the partial order≥ of knots.

## Questions about knots

By a knotK we mean a closed embedded curve inS^{3}.
We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(5) Determine which knots are periodic.

(6) Determine which knots are amphichiral.

(7) We writeK1 ≥K2 if there exists an epimorphism
π1(S^{3}\K1)→π1(S^{3}\K2).

This defines a partial order on the set of knots.

(7) Determine the partial order≥of knots.

## Questions about knots

By a knotK we mean a closed embedded curve inS^{3}.
We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(5) Determine which knots are periodic.

(6) Determine which knots are amphichiral.

(7) We writeK1 ≥K2 if there exists an epimorphism
π1(S^{3}\K1)→π1(S^{3}\K2).

This defines a partial order on the set of knots. Goal: determine the partial order of knots.

(7) Determine the partial order≥of knots.

## Questions about knots

By a knotK we mean a closed embedded curve inS^{3}.
We list some goals in knot theory.

(1) Find invariants which distinguish knots.

(2) Determine the genusg(K) ofK.

(5) Determine which knots are periodic.

(6) Determine which knots are amphichiral.

(7) Determine the partial order≥of knots.

## The classical Alexander polynomial of a knot: advanced definition

For a knotK we write X =S^{3}\K.

The classical Alexander polynomial of a knot: advanced definition

For a knotK we write X =S^{3}\K. We haveH1(S^{3}\K) =Zby
Alexander duality

The classical Alexander polynomial of a knot: advanced definition

For a knotK we write X =S^{3}\K. We haveH1(S^{3}\K) =Zby
Alexander duality and we denote by ˜X the infinite cyclic cover of X
corresponding toπ_{1}(X)→H_{1}(X)→Z=hti.

## The classical Alexander polynomial of a knot: advanced definition

For a knotK we write X =S^{3}\K. We haveH1(S^{3}\K) =Zby
Alexander duality and we denote by ˜X the infinite cyclic cover of X
corresponding toπ_{1}(X)→H_{1}(X)→Z=hti. The infinite cyclic
grouphti acts on H1( ˜X),

The classical Alexander polynomial of a knot: advanced definition

For a knotK we write X =S^{3}\K. We haveH1(S^{3}\K) =Zby
Alexander duality and we denote by ˜X the infinite cyclic cover of X
corresponding toπ_{1}(X)→H_{1}(X)→Z=hti. The infinite cyclic
grouphti acts on H1( ˜X), henceH1( ˜X) is a module overZ[t^{±1}].

The classical Alexander polynomial of a knot: advanced definition

For a knotK we write X =S^{3}\K. We haveH1(S^{3}\K) =Zby
Alexander duality and we denote by ˜X the infinite cyclic cover of X
corresponding toπ_{1}(X)→H_{1}(X)→Z=hti. The infinite cyclic
grouphti acts on H1( ˜X), henceH1( ˜X) is a module overZ[t^{±1}].

We write

H_{1}(X;Z[t^{±1}]) =H_{1}( ˜X).

## The classical Alexander polynomial of a knot: advanced definition

For a knotK we write X =S^{3}\K. We haveH1(S^{3}\K) =Zby
Alexander duality and we denote by ˜X the infinite cyclic cover of X
corresponding toπ_{1}(X)→H_{1}(X)→Z=hti. The infinite cyclic
grouphti acts on H1( ˜X), henceH1( ˜X) is a module overZ[t^{±1}].

We write

H_{1}(X;Z[t^{±1}]) =H_{1}( ˜X).

We have a resolution

Z[t^{±1}]^{n D}−→Z[t^{±1}]^{n}→H_{1}(X,Z[t^{±1}])→0
and we define

The classical Alexander polynomial of a knot: advanced definition

^{3}\K. We haveH1(S^{3}\K) =Zby
Alexander duality and we denote by ˜X the infinite cyclic cover of X
corresponding toπ_{1}(X)→H_{1}(X)→Z=hti. The infinite cyclic
grouphti acts on H1( ˜X), henceH1( ˜X) is a module overZ[t^{±1}].

We write

H_{1}(X;Z[t^{±1}]) =H_{1}( ˜X).

We have a resolution

Z[t^{±1}]^{n D}−→Z[t^{±1}]^{n}→H_{1}(X,Z[t^{±1}])→0
and we define

∆_{K}(t) = det(D)∈Z[t^{±1}].

## The classical Alexander polynomial of a knot: advanced definition

For a knotK we write X =S^{3}\K. We have a resolution
Z[t^{±1}]^{n D}−→Z[t^{±1}]^{n}→H_{1}(X,Z[t^{±1}])→0

and we define

∆_{K}(t) = det(D)∈Z[t^{±1}].

(1) IfAis a Seifert matrix, then D =At−A^{t} and hence

∆_{K}(t) = det(At −A^{t}).

The classical Alexander polynomial of a knot: advanced definition

For a knotK we write X =S^{3}\K. We have a resolution
Z[t^{±1}]^{n D}−→Z[t^{±1}]^{n}→H_{1}(X,Z[t^{±1}])→0

and we define

∆_{K}(t) = det(D)∈Z[t^{±1}].

(1) IfAis a Seifert matrix, then D =At−A^{t} and hence

∆_{K}(t) = det(At −A^{t}).

This approach is very effective for knots but does not generalize well to 3-manifolds.

## The classical Alexander polynomial of a knot: advanced definition

For a knotK we write X =S^{3}\K. We have a resolution
Z[t^{±1}]^{n D}−→Z[t^{±1}]^{n}→H_{1}(X,Z[t^{±1}])→0

and we define

∆_{K}(t) = det(D)∈Z[t^{±1}].

(1) IfAis a Seifert matrix, then D =At−A^{t} and hence

∆_{K}(t) = det(At −A^{t}).

(2) ∆_{K}(t) can be computed easily using Fox calculus.

The classical Alexander polynomial of a knot: advanced definition

^{3}\K. We have a resolution
Z[t^{±1}]^{n D}−→Z[t^{±1}]^{n}→H_{1}(X,Z[t^{±1}])→0

and we define

∆_{K}(t) = det(D)∈Z[t^{±1}].

(1) IfAis a Seifert matrix, then D =At−A^{t} and hence

∆_{K}(t) = det(At −A^{t}).

(2) ∆_{K}(t) can be computed easily using Fox calculus.

(3) ∆_{K}(t) can also be expressed using Reidemeister-Milnor-Turaev
torsion

## The classical Alexander polynomial of a knot: advanced definition

^{3}\K. We have a resolution
Z[t^{±1}]^{n D}−→Z[t^{±1}]^{n}→H_{1}(X,Z[t^{±1}])→0

and we define

∆_{K}(t) = det(D)∈Z[t^{±1}].

(1) IfAis a Seifert matrix, then D =At−A^{t} and hence

∆_{K}(t) = det(At −A^{t}).

(2) ∆_{K}(t) can be computed easily using Fox calculus.

(3) ∆_{K}(t) can also be expressed using Reidemeister-Milnor-Turaev
torsion (which is my favorite view point!)

Properties of the Alexander polynomial

LetK be a knot and ∆_{K}(t) its Alexander polynomial.

(1) ∆_{K}(t)∈Z[t^{±1}] (2) The Alexander polynomial of the trivial
knot equals 1.

(3) There are non-trivial knots with Alexander polynomial 1
(4) The Alexander polynomial is unchanged under mutation.
(5) ∆_{K}(t) = ∆_{K}(t^{−1})

(6) ∆_{K}(1) =±1

(7) deg(∆_{K}(t))≤2g(K)

(8) IfK is fibered, then deg(∆_{K}(t))≤2g(K) and ∆_{K}(t) is
monic.

(9) IfK is slice, then ∆_{K}(t) =f(t)f(t^{−1}) for some f(t)∈Z[t^{±1}]
(10) ∆_{K}^{∗}(t) = ∆_{K}(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

## Properties of the Alexander polynomial

LetK be a knot and ∆_{K}(t) its Alexander polynomial.

(1) ∆_{K}(t)∈Z[t^{±1}]

(2) The Alexander polynomial of the trivial knot equals 1.

(3) There are non-trivial knots with Alexander polynomial 1
(4) The Alexander polynomial is unchanged under mutation.
(5) ∆_{K}(t) = ∆_{K}(t^{−1})

(6) ∆_{K}(1) =±1

(7) deg(∆_{K}(t))≤2g(K)

(8) IfK is fibered, then deg(∆_{K}(t))≤2g(K) and ∆_{K}(t) is
monic.

(9) IfK is slice, then ∆_{K}(t) =f(t)f(t^{−1}) for some f(t)∈Z[t^{±1}]
(10) ∆_{K}^{∗}(t) = ∆_{K}(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

## Properties of the Alexander polynomial

LetK be a knot and ∆_{K}(t) its Alexander polynomial.

(1) ∆_{K}(t)∈Z[t^{±1}] and is well-defined up to
multiplication by±t^{k}.

(2) The Alexander polynomial of the trivial knot equals 1.
(3) There are non-trivial knots with Alexander polynomial 1
(4) The Alexander polynomial is unchanged under mutation.
(5) ∆_{K}(t) = ∆_{K}(t^{−1})

(6) ∆_{K}(1) =±1

(7) deg(∆_{K}(t))≤2g(K)

(8) IfK is fibered, then deg(∆_{K}(t))≤2g(K) and ∆_{K}(t) is
monic.

(9) IfK is slice, then ∆_{K}(t) =f(t)f(t^{−1}) for some f(t)∈Z[t^{±1}]
(10) ∆K^{∗}(t) = ∆K(t)

(11) IfK_{1} ≥K_{2}, then ∆_{K}_{2}(t) divides ∆_{K}_{1}(t)

(12) The Alexander polynomial of a periodic knot has a special form

## Properties of the Alexander polynomial

LetK be a knot and ∆_{K}(t) its Alexander polynomial.

(1) ∆_{K}(t)∈Z[t^{±1}] and well-def. up to multiplication by±t^{k}.
(2) The Alexander polynomial of the trivial knot equals 1.

(3) There are non-trivial knots with Alexander polynomial 1
(4) The Alexander polynomial is unchanged under mutation.
(5) ∆_{K}(t) = ∆_{K}(t^{−1})

(6) ∆_{K}(1) =±1

(7) deg(∆_{K}(t))≤2g(K)

(8) IfK is fibered, then deg(∆_{K}(t))≤2g(K) and ∆_{K}(t) is
monic.

(9) IfK is slice, then ∆_{K}(t) =f(t)f(t^{−1}) for some f(t)∈Z[t^{±1}]
(10) ∆_{K}^{∗}(t) = ∆_{K}(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

## Properties of the Alexander polynomial

LetK be a knot and ∆_{K}(t) its Alexander polynomial.

(1) ∆_{K}(t)∈Z[t^{±1}] and well-def. up to multiplication by±t^{k}.
(2) The Alexander polynomial of the trivial knot equals 1.

The Alexander polynomial of the trefoil knot equalst^{−1}−1 +t.

_{K}(t) = ∆_{K}(t^{−1})

(6) ∆_{K}(1) =±1

(7) deg(∆_{K}(t))≤2g(K)

(8) IfK is fibered, then deg(∆_{K}(t))≤2g(K) and ∆_{K}(t) is
monic.

(9) IfK is slice, then ∆_{K}(t) =f(t)f(t^{−1}) for some f(t)∈Z[t^{±1}]
(10) ∆K^{∗}(t) = ∆K(t)

(11) IfK_{1} ≥K_{2}, then ∆_{K}_{2}(t) divides ∆_{K}_{1}(t)

(12) The Alexander polynomial of a periodic knot has a special form

## Properties of the Alexander polynomial

LetK be a knot and ∆_{K}(t) its Alexander polynomial.

(1) ∆_{K}(t)∈Z[t^{±1}] and well-def. up to multiplication by±t^{k}.
(2) The Alexander polynomial of the trivial knot equals 1.

(3) There are non-trivial knots with Alexander polynomial 1

(4) The Alexander polynomial is unchanged under mutation.
(5) ∆_{K}(t) = ∆_{K}(t^{−1})

(6) ∆_{K}(1) =±1

(7) deg(∆_{K}(t))≤2g(K)

(8) IfK is fibered, then deg(∆_{K}(t))≤2g(K) and ∆_{K}(t) is
monic.

_{K}(t) =f(t)f(t^{−1}) for some f(t)∈Z[t^{±1}]
(10) ∆_{K}^{∗}(t) = ∆_{K}(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

## Properties of the Alexander polynomial

LetK be a knot and ∆_{K}(t) its Alexander polynomial.

_{K}(t)∈Z[t^{±1}] and well-def. up to multiplication by±t^{k}.
(2) The Alexander polynomial of the trivial knot equals 1.

(3) There are non-trivial knots with Alexander polynomial 1 so the Alexander polynomial is not a complete invariant of knots

(4) The Alexander polynomial is unchanged under mutation.
(5) ∆_{K}(t) = ∆_{K}(t^{−1})

(6) ∆_{K}(1) =±1

(7) deg(∆_{K}(t))≤2g(K)

(8) IfK is fibered, then deg(∆_{K}(t))≤2g(K) and ∆_{K}(t) is
monic.

(9) IfK is slice, then ∆_{K}(t) =f(t)f(t^{−1}) for some f(t)∈Z[t^{±1}]
(10) ∆K^{∗}(t) = ∆K(t)

(11) IfK_{1} ≥K_{2}, then ∆_{K}_{2}(t) divides ∆_{K}_{1}(t)

(12) The Alexander polynomial of a periodic knot has a special form

## Properties of the Alexander polynomial

LetK be a knot and ∆_{K}(t) its Alexander polynomial.

_{K}(t)∈Z[t^{±1}] and well-def. up to multiplication by±t^{k}.
(2) The Alexander polynomial of the trivial knot equals 1.

(3) There are non-trivial knots with Alexander polynomial 1 (4) The Alexander polynomial is unchanged under mutation.

(5) ∆_{K}(t) = ∆_{K}(t^{−1})
(6) ∆_{K}(1) =±1

(7) deg(∆_{K}(t))≤2g(K)

(8) IfK is fibered, then deg(∆_{K}(t))≤2g(K) and ∆_{K}(t) is
monic.

_{K}(t) =f(t)f(t^{−1}) for some f(t)∈Z[t^{±1}]
(10) ∆_{K}^{∗}(t) = ∆_{K}(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

## Properties of the Alexander polynomial

LetK be a knot and ∆_{K}(t) its Alexander polynomial.

_{K}(t)∈Z[t^{±1}] and well-def. up to multiplication by±t^{k}.
(2) The Alexander polynomial of the trivial knot equals 1.

(3) There are non-trivial knots with Alexander polynomial 1 (4) The Alexander polynomial is unchanged under mutation.

(5) ∆K(t) = ∆K(t^{−1})

(6) ∆_{K}(1) =±1

(7) deg(∆_{K}(t))≤2g(K)

(8) IfK is fibered, then deg(∆_{K}(t))≤2g(K) and ∆_{K}(t) is
monic.

_{K}(t) =f(t)f(t^{−1}) for some f(t)∈Z[t^{±1}]
(10) ∆_{K}^{∗}(t) = ∆_{K}(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

## Properties of the Alexander polynomial

LetK be a knot and ∆_{K}(t) its Alexander polynomial.

_{K}(t)∈Z[t^{±1}] and well-def. up to multiplication by±t^{k}.
(2) The Alexander polynomial of the trivial knot equals 1.

(3) There are non-trivial knots with Alexander polynomial 1 (4) The Alexander polynomial is unchanged under mutation.

(5) ∆K(t) = ∆K(t^{−1})

(this is a consequence of Poincar´e duality)

(6) ∆_{K}(1) =±1
(7) deg(∆_{K}(t))≤2g(K)

(8) IfK is fibered, then deg(∆_{K}(t))≤2g(K) and ∆_{K}(t) is
monic.

_{K}(t) =f(t)f(t^{−1}) for some f(t)∈Z[t^{±1}]
(10) ∆_{K}^{∗}(t) = ∆_{K}(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

## Properties of the Alexander polynomial

LetK be a knot and ∆_{K}(t) its Alexander polynomial.

_{K}(t)∈Z[t^{±1}] and well-def. up to multiplication by±t^{k}.
(2) The Alexander polynomial of the trivial knot equals 1.

(5) ∆K(t) = ∆K(t^{−1})
(6) ∆_{K}(1) =±1

(7) deg(∆_{K}(t))≤2g(K)

(8) IfK is fibered, then deg(∆_{K}(t))≤2g(K) and ∆_{K}(t) is
monic.

_{K}(t) =f(t)f(t^{−1}) for some f(t)∈Z[t^{±1}]
(10) ∆_{K}^{∗}(t) = ∆_{K}(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

## Properties of the Alexander polynomial

LetK be a knot and ∆_{K}(t) its Alexander polynomial.

_{K}(t)∈Z[t^{±1}] and well-def. up to multiplication by±t^{k}.
(2) The Alexander polynomial of the trivial knot equals 1.

(5) ∆K(t) = ∆K(t^{−1})
(6) ∆_{K}(1) =±1

(ForK a null-homologous knot in a homology sphere Σ we have

∆K(1) =|H_{1}(Σ)|)

(7) deg(∆_{K}(t))≤2g(K)

(8) IfK is fibered, then deg(∆_{K}(t))≤2g(K) and ∆_{K}(t) is
monic.

_{K}(t) =f(t)f(t^{−1}) for some f(t)∈Z[t^{±1}]
(10) ∆K^{∗}(t) = ∆K(t)

(11) IfK_{1} ≥K_{2}, then ∆_{K}_{2}(t) divides ∆_{K}_{1}(t)

(12) The Alexander polynomial of a periodic knot has a special form

## Properties of the Alexander polynomial

LetK be a knot and ∆_{K}(t) its Alexander polynomial.

_{K}(t)∈Z[t^{±1}] and well-def. up to multiplication by±t^{k}.
(2) The Alexander polynomial of the trivial knot equals 1.

(5) ∆K(t) = ∆K(t^{−1})
(6) ∆_{K}(1) =±1

(7) deg(∆_{K}(t))≤2g(K)

(8) IfK is fibered, then deg(∆_{K}(t))≤2g(K) and ∆_{K}(t) is
monic.

_{K}(t) =f(t)f(t^{−1}) for some f(t)∈Z[t^{±1}]
(10) ∆_{K}^{∗}(t) = ∆_{K}(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

## Properties of the Alexander polynomial

LetK be a knot and ∆_{K}(t) its Alexander polynomial.

_{K}(t)∈Z[t^{±1}] and well-def. up to multiplication by±t^{k}.
(2) The Alexander polynomial of the trivial knot equals 1.

(5) ∆K(t) = ∆K(t^{−1})
(6) ∆_{K}(1) =±1

(7) deg(∆_{K}(t))≤2g(K)

(This is a consequence of ∆K(t) = det(At−A^{t}) whereAcan be a
Seifert matrix of size 2g(K)×2g(K)).

(8) IfK is fibered, then deg(∆_{K}(t))≤2g(K) and ∆_{K}(t) is
monic.

(9) IfK is slice, then ∆K(t) =f(t)f(t^{−1}) for some f(t)∈Z[t^{±1}]
(10) ∆_{K}^{∗}(t) = ∆_{K}(t)

(11) IfK_{1} ≥K_{2}, then ∆_{K}_{2}(t) divides ∆_{K}_{1}(t)

(12) The Alexander polynomial of a periodic knot has a special form

## Properties of the Alexander polynomial

LetK be a knot and ∆_{K}(t) its Alexander polynomial.

_{K}(t)∈Z[t^{±1}] and well-def. up to multiplication by±t^{k}.
(2) The Alexander polynomial of the trivial knot equals 1.

(5) ∆K(t) = ∆K(t^{−1})
(6) ∆_{K}(1) =±1

(7) deg(∆_{K}(t))≤2g(K)

(8) IfK is fibered, then deg(∆K(t))≤2g(K)

and ∆_{K}(t) is
monic.

_{K}(t) =f(t)f(t^{−1}) for some f(t)∈Z[t^{±1}]
(10) ∆_{K}^{∗}(t) = ∆_{K}(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

## Properties of the Alexander polynomial

LetK be a knot and ∆_{K}(t) its Alexander polynomial.

_{K}(t)∈Z[t^{±1}] and well-def. up to multiplication by±t^{k}.
(2) The Alexander polynomial of the trivial knot equals 1.

(5) ∆K(t) = ∆K(t^{−1})
(6) ∆_{K}(1) =±1

(7) deg(∆_{K}(t))≤2g(K)

(8) IfK is fibered, then deg(∆K(t))≤2g(K) and ∆K(t) is monic i.e. the top coefficient is±1.

and ∆_{K}(t) is monic.

_{K}(t) =f(t)f(t^{−1}) for some f(t)∈Z[t^{±1}]
(10) ∆K^{∗}(t) = ∆K(t)

(11) IfK_{1} ≥K_{2}, then ∆_{K}_{2}(t) divides ∆_{K}_{1}(t)

(12) The Alexander polynomial of a periodic knot has a special form

## Properties of the Alexander polynomial

LetK be a knot and ∆_{K}(t) its Alexander polynomial.

_{K}(t)∈Z[t^{±1}] and well-def. up to multiplication by±t^{k}.
(2) The Alexander polynomial of the trivial knot equals 1.

(5) ∆K(t) = ∆K(t^{−1})
(6) ∆_{K}(1) =±1

(7) deg(∆_{K}(t))≤2g(K)

(8) IfK is fibered, then deg(∆K(t))≤2g(K) and ∆K(t) is monic i.e. the top coefficient is±1.

and ∆_{K}(t) is monic.

(IfK is fibered andA a Seifert matrix for a fiber, then det(A) = 1,

(9) IfK is slice, then ∆K(t) =f(t)f(t^{−1}) for somef(t)∈Z[t^{±1}]
(10) ∆_{K}^{∗}(t) = ∆_{K}(t)

(11) IfK_{1} ≥K_{2}, then ∆_{K}_{2}(t) divides ∆_{K}_{1}(t)

(12) The Alexander polynomial of a periodic knot has a special form

## Properties of the Alexander polynomial

LetK be a knot and ∆_{K}(t) its Alexander polynomial.

_{K}(t)∈Z[t^{±1}] and well-def. up to multiplication by±t^{k}.
(2) The Alexander polynomial of the trivial knot equals 1.

(5) ∆K(t) = ∆K(t^{−1})
(6) ∆_{K}(1) =±1

(7) deg(∆_{K}(t))≤2g(K)

(8) IfK is fibered, then deg(∆K(t))≤2g(K) and ∆K(t) is monic i.e. the top coefficient is±1.

and ∆_{K}(t) is monic.

(IfK is fibered andA a Seifert matrix for a fiber, then det(A) = 1,
so the claim follows from ∆_{K}(t) = det(At−A^{t})).

_{K}(t) =f(t)f(t^{−1}) for some f(t)∈Z[t^{±1}]
(10) ∆_{K}^{∗}(t) = ∆_{K}(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

## Properties of the Alexander polynomial

LetK be a knot and ∆_{K}(t) its Alexander polynomial.

_{K}(t)∈Z[t^{±1}] and well-def. up to multiplication by±t^{k}.
(2) The Alexander polynomial of the trivial knot equals 1.

(5) ∆K(t) = ∆K(t^{−1})
(6) ∆_{K}(1) =±1

(7) deg(∆_{K}(t))≤2g(K)

(8) IfK is fibered, then deg(∆K(t))≤2g(K) and ∆K(t) is monic.

(9) IfK is slice, then ∆_{K}(t) =f(t)f(t^{−1}) for some f(t)∈Z[t^{±1}]

(10) ∆_{K}^{∗}(t) = ∆_{K}(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

## Properties of the Alexander polynomial

LetK be a knot and ∆_{K}(t) its Alexander polynomial.

_{K}(t)∈Z[t^{±1}] and well-def. up to multiplication by±t^{k}.
(2) The Alexander polynomial of the trivial knot equals 1.

(5) ∆K(t) = ∆K(t^{−1})
(6) ∆_{K}(1) =±1

(7) deg(∆_{K}(t))≤2g(K)

(8) IfK is fibered, then deg(∆K(t))≤2g(K) and ∆K(t) is monic.

(9) IfK is slice, then ∆_{K}(t) =f(t)f(t^{−1}) for some f(t)∈Z[t^{±1}]
(IfD⊂D^{4} is a slice disk, this follows from Poincar´e duality
applied to the pair (D^{4}\D,S^{3}\K))

(10) ∆_{K}^{∗}(t) = ∆_{K}(t)

(11) IfK_{1} ≥K_{2}, then ∆_{K}_{2}(t) divides ∆_{K}_{1}(t)

(12) The Alexander polynomial of a periodic knot has a special form

## Properties of the Alexander polynomial

LetK be a knot and ∆_{K}(t) its Alexander polynomial.

_{K}(t)∈Z[t^{±1}] and well-def. up to multiplication by±t^{k}.
(2) The Alexander polynomial of the trivial knot equals 1.

(5) ∆K(t) = ∆K(t^{−1})
(6) ∆_{K}(1) =±1

(7) deg(∆_{K}(t))≤2g(K)

(8) IfK is fibered, then deg(∆K(t))≤2g(K) and ∆K(t) is monic.

_{K}(t) =f(t)f(t^{−1}) for some f(t)∈Z[t^{±1}]
(10) ∆_{K}^{∗}(t) = ∆_{K}(t)

(11) IfK1 ≥K2, then ∆K2(t) divides ∆K1(t)

(12) The Alexander polynomial of a periodic knot has a special form

## Properties of the Alexander polynomial

LetK be a knot and ∆_{K}(t) its Alexander polynomial.

_{K}(t)∈Z[t^{±1}] and well-def. up to multiplication by±t^{k}.
(2) The Alexander polynomial of the trivial knot equals 1.

(5) ∆K(t) = ∆K(t^{−1})
(6) ∆_{K}(1) =±1

(7) deg(∆_{K}(t))≤2g(K)

(8) IfK is fibered, then deg(∆K(t))≤2g(K) and ∆K(t) is monic.

_{K}(t) =f(t)f(t^{−1}) for some f(t)∈Z[t^{±1}]
(10) ∆_{K}^{∗}(t) = ∆_{K}(t)

i.e. the Alexander polynomial does not distinguish between mirror images

(11) IfK_{1} ≥K_{2}, then ∆_{K}_{2}(t) divides ∆_{K}_{1}(t)

(12) The Alexander polynomial of a periodic knot has a special form

## Properties of the Alexander polynomial

LetK be a knot and ∆_{K}(t) its Alexander polynomial.

_{K}(t)∈Z[t^{±1}] and well-def. up to multiplication by±t^{k}.
(2) The Alexander polynomial of the trivial knot equals 1.

(5) ∆K(t) = ∆K(t^{−1})
(6) ∆_{K}(1) =±1

(7) deg(∆_{K}(t))≤2g(K)

(8) IfK is fibered, then deg(∆K(t))≤2g(K) and ∆K(t) is monic.

_{K}(t) =f(t)f(t^{−1}) for some f(t)∈Z[t^{±1}]
(10) ∆_{K}^{∗}(t) = ∆_{K}(t)

(11) IfK1 ≥K2, then ∆_{K}_{2}(t) divides ∆_{K}_{1}(t)

(12) The Alexander polynomial of a periodic knot has a special form

## Properties of the Alexander polynomial

LetK be a knot and ∆_{K}(t) its Alexander polynomial.

_{K}(t)∈Z[t^{±1}] and well-def. up to multiplication by±t^{k}.
(2) The Alexander polynomial of the trivial knot equals 1.

(5) ∆K(t) = ∆K(t^{−1})
(6) ∆_{K}(1) =±1

(7) deg(∆_{K}(t))≤2g(K)

(8) IfK is fibered, then deg(∆K(t))≤2g(K) and ∆K(t) is monic.

_{K}(t) =f(t)f(t^{−1}) for some f(t)∈Z[t^{±1}]
(10) ∆_{K}^{∗}(t) = ∆_{K}(t)

(11) IfK1 ≥K2, then ∆_{K}_{2}(t) divides ∆_{K}_{1}(t)

(12) The Alexander polynomial of a periodic knot has a special form

## Twisted Alexander polynomials: homological definition

LetK ⊂S^{3} and α:π=π1(S^{3}\K)→GL(n,R) a representation
over a UFD

Denote the epimorphism π →Zbyφand the
universal cover ofX =S^{3}\K by ˜X. Z[π] acts onC∗( ˜X) by deck
transformations andZ[π] acts onR[t^{±1}]⊗R^{n}=R^{n}[t^{±1}]

Twisted Alexander polynomials: homological definition

LetK ⊂S^{3} and α:π=π1(S^{3}\K)→GL(n,R) a representation
over a UFD (e.g. Zor C)

Denote the epimorphism π→Zbyφ
and the universal cover ofX =S^{3}\K by ˜X. Z[π] acts onC∗( ˜X)
by deck transformations andZ[π] acts onR[t^{±1}]⊗R^{n}=R^{n}[t^{±1}]

## Twisted Alexander polynomials: homological definition

LetK ⊂S^{3} and α:π=π1(S^{3}\K)→GL(n,R) a representation
over a UFD Denote the epimorphismπ →Zbyφ

and the
universal cover ofX =S^{3}\K by ˜X. Z[π] acts onC∗( ˜X) by deck
transformations andZ[π] acts onR[t^{±1}]⊗R^{n}=R^{n}[t^{±1}]

Twisted Alexander polynomials: homological definition

LetK ⊂S^{3} and α:π=π1(S^{3}\K)→GL(n,R) a representation
over a UFD Denote the epimorphismπ →Zbyφand the
universal cover ofX =S^{3}\K by ˜X.

Z[π] acts onC∗( ˜X) by deck
transformations andZ[π] acts onR[t^{±1}]⊗R^{n}=R^{n}[t^{±1}]

## Twisted Alexander polynomials: homological definition

LetK ⊂S^{3} and α:π=π1(S^{3}\K)→GL(n,R) a representation
over a UFD Denote the epimorphismπ →Zbyφand the
universal cover ofX =S^{3}\K by ˜X. Z[π] acts onC∗( ˜X) by deck
transformations

andZ[π] acts onR[t^{±1}]⊗R^{n}=R^{n}[t^{±1}]

Twisted Alexander polynomials: homological definition

LetK ⊂S^{3} and α:π=π1(S^{3}\K)→GL(n,R) a representation
over a UFD Denote the epimorphismπ →Zbyφand the
universal cover ofX =S^{3}\K by ˜X. Z[π] acts onC∗( ˜X) by deck
transformations andZ[π] acts onR[t^{±1}]⊗R^{n}=R^{n}[t^{±1}] as
follows:

## Twisted Alexander polynomials: homological definition

LetK ⊂S^{3} and α:π=π1(S^{3}\K)→GL(n,R) a representation
over a UFD Denote the epimorphismπ →Zbyφand the
universal cover ofX =S^{3}\K by ˜X. Z[π] acts onC∗( ˜X) by deck
transformations andZ[π] acts onR[t^{±1}]⊗R^{n}=R^{n}[t^{±1}] as
follows:

g·(p(t)⊗v) =t^{φ(g)}p(t)⊗α(g)v.

Twisted Alexander polynomials: homological definition

LetK ⊂S^{3} and α:π=π1(S^{3}\K)→GL(n,R) a representation
over a UFD Denote the epimorphismπ →Zbyφand the
universal cover ofX =S^{3}\K by ˜X. Z[π] acts onC∗( ˜X) by deck
transformations andZ[π] acts onR[t^{±1}]⊗R^{n}=R^{n}[t^{±1}] as
follows:

g·(p(t)⊗v) =t^{φ(g)}p(t)⊗α(g)v.

Consider C_{∗}^{α}(X;R^{n}[t^{±1}]) :=C∗( ˜X)⊗_{Z}_{[π]}R^{n}[t^{±1}]

## Twisted Alexander polynomials: homological definition

^{3} and α:π=π1(S^{3}\K)→GL(n,R) a representation
over a UFD Denote the epimorphismπ →Zbyφand the
universal cover ofX =S^{3}\K by ˜X. Z[π] acts onC∗( ˜X) by deck
transformations andZ[π] acts onR[t^{±1}]⊗R^{n}=R^{n}[t^{±1}] as
follows:

g·(p(t)⊗v) =t^{φ(g)}p(t)⊗α(g)v.

Consider C_{∗}^{α}(X;R^{n}[t^{±1}]) :=C∗( ˜X)⊗_{Z}_{[π]}R^{n}[t^{±1}]
(this is a chain complex over the ringR[t^{±1}])

Twisted Alexander polynomials: homological definition

^{3} and α:π=π1(S^{3}\K)→GL(n,R) a representation
over a UFD Denote the epimorphismπ →Zbyφand the
universal cover ofX =S^{3}\K by ˜X. Z[π] acts onC∗( ˜X) by deck
transformations andZ[π] acts onR[t^{±1}]⊗R^{n}=R^{n}[t^{±1}] as
follows:

g·(p(t)⊗v) =t^{φ(g)}p(t)⊗α(g)v.

Consider C_{∗}^{α}(X;R^{n}[t^{±1}]) :=C∗( ˜X)⊗_{Z}_{[π]}R^{n}[t^{±1}]

(this is a chain complex over the ringR[t^{±1}]) and its homology
H_{∗}^{α}(X;R^{n}[t^{±1}]).

## Twisted Alexander polynomials: homological definition

^{3} and α:π=π1(S^{3}\K)→GL(n,R) a representation
over a UFD Denote the epimorphismπ →Zbyφand the
universal cover ofX =S^{3}\K by ˜X. Z[π] acts onC∗( ˜X) by deck
transformations andZ[π] acts onR[t^{±1}]⊗R^{n}=R^{n}[t^{±1}] as
follows:

g·(p(t)⊗v) =t^{φ(g)}p(t)⊗α(g)v.

Consider C_{∗}^{α}(X;R^{n}[t^{±1}]) :=C∗( ˜X)⊗_{Z}_{[π]}R^{n}[t^{±1}]

(this is a chain complex over the ringR[t^{±1}]) and its homology
H_{∗}^{α}(X;R^{n}[t^{±1}]). Pick a resolution

R^{n}[t^{±1}]^{k D}−→R^{n}[t^{±1}]^{l} →H_{∗}^{α}(X;R^{n}[t^{±1}])→0

Twisted Alexander polynomials: homological definition

^{3} and α:π=π1(S^{3}\K)→GL(n,R) a representation
over a UFD Denote the epimorphismπ →Zbyφand the
universal cover ofX =S^{3}\K by ˜X. Z[π] acts onC∗( ˜X) by deck
transformations andZ[π] acts onR[t^{±1}]⊗R^{n}=R^{n}[t^{±1}] as
follows:

g·(p(t)⊗v) =t^{φ(g)}p(t)⊗α(g)v.

Consider C_{∗}^{α}(X;R^{n}[t^{±1}]) :=C∗( ˜X)⊗_{Z}_{[π]}R^{n}[t^{±1}]

(this is a chain complex over the ringR[t^{±1}]) and its homology
H_{∗}^{α}(X;R^{n}[t^{±1}]). Pick a resolution

R^{n}[t^{±1}]^{k D}−→R^{n}[t^{±1}]^{l} →H_{∗}^{α}(X;R^{n}[t^{±1}